Multi-Step stochastic correction in dynamical fermion updating algorithms

The advantages of using Multi-Step corrections for simulations of lattice gauge theories with dynamical fermions will be discussed. This technique is suited for algorithms based on the Multi-Boson representation of the dynamical fermions as well as for the Hybrid Monte-Carlo (HMC) algorithm and variants of the latter, like the Polynomial-HMC. Especially the latter has the power to deal with an odd number of fermion fields--an essential feature necessary for realistic QCD-simulations with up-, down-, and strange-quarks. In particular, we will discuss the application of the multi-step (actually two-step) correction technique to a PHMC updating algorithm for twisted-mass Wilson fermions with non-degenerate fermion masses, as it was used in recent dynamical simulations for N_f=2+1+1 fermion flavors.